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L stecag ojgto arxi the in matrix conjugation charge the is n D → a hsc Institute, Physics cal T ,4 3, Cγ and = m e | Aul,Italy ’Aquila, ∆ iα − 5 i dQakOperators Quark nd B| , n hc sra n oiie The positive. and real is which nγ P n, B i m 2 e rcse and, processes 2 = µ = ∂ = nγ µ nC n − n − 5 → for 1 , n C ,isedof instead 1, − C n n , e T m = α − P , , iα nn − n and ( B t conservation Its . n γ nCγ α 0 symmetry . 1 = T n 5 n n ¯ ..., , T → . e )and 4) iηγ 5 (2) (3) (1) (4) n . 2 to just one possibility for the baryon charge breaking by eigenvalues of P are 1 and opposite for fermion and two units, antifermion states. ± Different parities of neutron and antineutron imply 1 T T P ∆ 6B = ǫ n Cn + nCn , (5) that their mixing breaks parity, and, indeed, the sub- L −2 stitution (11) changes ∆ 6B to ( ∆ 6B) . Together with   L − L where ǫ is a real positive parameter. The possibility C invariance it implies then that ∆ 6B is also CP odd. L of such redefinitions is based on U(2) symmetry of the However, this CP oddness does not translate immedi- µ CP kinetic term i nγ ∂µn. Four-parametric U(2) transfor- ately into observable breaking effects. To get them mations allow to exclude the term (3) and to reduce four one needs an interference of amplitudes and this is pro- terms (4) to just one structure (5). vided only when interaction is present. It shows a subtlety in the definition of parity transfor- P 3. What is the status of discrete C, P and T symmetries mation , see textbook discussions, e.g., in Refs. [11, 12]. under the baryon charge breaking modification (5)? Let Let us remind it. us first consider the charge conjugation C, which can be When baryon charge is conserved there is no transition P viewed as a plain exchange symmetry between n and nc between sectors with different , and one can combine B P fields, with a baryonic U(1)B phase rotation (2) and define α,

c T iBα iα 0 c −iα 0 c C : n n = C n . (6) Pα = P e : n e γ n, n e γ n . (12) ←→ → →− C2 P2 2iBα This is a sort of discrete Z2 symmetry, = 1. The most Of course, then α = e = 1 but the phase is unob- simple it looks in the Majorana representation where servable when is conserved.6 When baryonB charge is not conserved the only rem- nc = n∗ . (7) nant of baryonic U(1)B rotations is Z2 symmetry asso- ciated with changing sign of the fermion field, n n. It is straithforward to verify that both Lagrangians → − above, (1) and (5), are C invariant. Indeed, they could This symmetry is protected: unphysical 2π space rota- be rewritten in the form tion changes the sign of the fermion field. It means that besides the original P 2 = 1 we can consider a different i µ c µ c m c c P P2 = nγ ∂ n + n γ ∂ n nn + n n , parity definition z , such that z = 1. LD 2 µ µ − 2 −     (8) Thus, choosing α = π/2 in Eq.(12), we come to a new ǫ c c P ∆ 6B = n n + nn , parity z, L −2   C P = P eiBπ/2 : n iγ0n, nc iγ0nc (13) which makes their invariance explicit. z → → The Lagrangians are diagonalized in terms of Majo- with P2 = 1. Now P parities of n and nc states are rana fields n1,2 , z z the same and− equal to i, so their mixing does not break c n n Pz parity. It means that all discrete symmetries, C, Pz n1,2 = ± , (9) T √2 and are preserved by the baryon breaking term ∆ 6B . Couple of related comments. First, one can chooseL which are even and odd under the charge conjugation C, c α = π/2 and have parities of fermion and antifermion n = n1 2. Namely, − 1,2 ± , both equal to ( i) instead of i. The absolute sign has no physical meaning− – it could be changed by a 2π 1 µ D = nkγ ∂µnk m nknk , space rotation – but relative parity between two dif- L 2 − kX=1,2   (10) ferent fermions does make sense. Second, it is amus- c 1 ing that the same Pz parity for n and n equal to i ∆ 6B = ǫ n1 n1 n2 n2 . L −2 − is still consistent with the notion of opposite parities of   fermion and antifermion, having in mind that that for It demonstrates that the baryon charge breaking leads to the complex value of parity we should compare Pz(n) splitting into two Majorana spin doublets. The C-even c ∗ with [Pz(n )] . Also for a fermion-antifermion pair the n1 field gets the mass M1 = m + ǫ while the mass of the c product Pz(n)Pz(n ) = 1. One more comment is to C-odd n2 is M2 = m ǫ. P − C CP P C − P notice that z commutes with , i.e., z= z , in Turn now to the parity transformation . It involves contrast with P which instead anticommutes with C, i.e., (besides reflection of the space coordinates) the substitu- CP= PC . For Majorana fermion both charge and par- tion ity conjugations− are diagonal in the Hilbert space: their P : n γ0n, nc γ0nc , (11) actions (in the rest frame) do not lead to a different phys- → →− ical state. It means that only the commuting case, i.e., 0 0 where γ Cγ = C is used. The opposite signs in Pz not P, is allowed. transformations for−n and nc reflect the well-known the- Thus, we demonstrated that neutron-antineutron mix- orem [10] on the opposite parities of fermion and an- ing by ∆ = 2 Majorana term in the mass matrix leads 2 B ± tifermion. The definition (11) satisfies P = 1, so the to a specific definition of the conserved parity Pz, making 3 it complex and satisfying P2 = 1 instead of (+1). It have: z − is this definition which should be used in analyzing CPz 1 1α 2α ψ ψ c ψ violating interactions. n = 2 2 ∗ = , n = . (16)  iσ (ψ )   ψ¯2˙   ψ¯1˙  Having in mind that invariance under the charge con- − α α jugation was already checked, preservation of T invari- The generic Lorentz invariant Lagrangian quadratic in CPT iα ance follows from theorem provided by Lorentz in- fermionic fields ψ and ψiα˙ is variance and locality. A specific Pz definition of parity T i k αα˙ α˙ kα transformation defines a specific transformation. = ψ ∂ ψ + ψ ∂ ˙ ψ L 2 α kα˙ k αα A few words about the history of the parity definition.   (17) As we mentioned earlier Ettore Majorana and Giulio 1 i kα ki α˙ P2 mik ψα ψ + m ψk ψiα˙ , Racah were the first to realize a necessity of z = 1 − 2   in application to Majorana fermions [8, 9]. The case− of where ∂αα˙ =(σµ)αα˙ ∂ , σµ = 1, ~σ , and ∂ =(¯σ µ) ∂ , neutron-antineutron mixing is essentially the same be- µ αα˙ αα˙ µ σ¯µ = 1, ~σ , m is the symmetric{ } mass matrix, m = cause it leads to splitting into two Majorana fermions of ik ik m and{ −mik} = (m )∗ is its conjugate. different mass and opposite C-parities. Years later this ki ik In the above equation we are implying a standard di- definition of parity was applied to Majorana neutrino in agonal form for kinetic terms. These terms in (17) are Refs. [13]. U(2) symmetric: besides flavor SU(2) rotations it in- Let us now comment on the recent applications cludes also U(1) associated with the overall phase ro- [5–7] to neutron-antineutron oscillations. Fujikawa tation of the flavor doublet (14) which in terms of Dirac and Tureanu in [5] came to incorrect conclusion about spinors (16) is just a chiral transformation. The U(2) necessity of P-parity breaking in ∆ = 2 processes. B ± P2 symmetry of kinetic terms it clearly generic: starting Similar to our initial claim in [4], it is due = 1 for α˙ i kα i the parity definition what leads to the opposite parities with i ψi Ck ∂αα˙ ψ where Ck is an arbitrary Hermitian 1 for neutron and antineutron. McKeen and Nelson flavor matrix, one can always diagonalize and normalize in± [6] also missed this point, and, as we mentioned at these terms. the beginning, incorrectly insisted that one can stay As for the mass terms they generically break both, with P2 = 1. Technically, the origin of the mistake U(1) and SU(2) flavor symmetries, so no continuous sym- is that their ∆ = 2 Lagrangian, given by Eq.(A6) metry remains. To see how the U(1)B symmetry (2) as- in [6], becomesB a total± derivative when C and P are sociated with the baryon charge could survive note that conserved with P2 = 1. Then, all its matrix elements one can interpret U(2) transformations as acting on the vanish - no oscillations. Gardner and Yan in [7] followed external mass matrix mik. This matrix is charged under Majorana neutrino case [13] and correctly defined the U(1), the overall phase rotation, so this U(1) symmetry P2 P is always broken by nonvanishing mass. In respect to parity inversion with z = 1 and the same z = i for botn, neutron and antineutron.− SU(2) transformations the symmetric tensor mik is the adjoint representation, i.e., can be viewed as an isovector µa, a =1, 2, 3, 4. To show that the above consideration covers a generic i ij a a i case it is convenient to introduce two left-handed Weyl mk = ε mjk = µ (τ )k , a =1, 2, 3 , (18) spinors, forming a flavor doublet1 Because µa is complex, we are actually dealing with two real isovectors, Re µa and Im µa. The SU(2) transforma- ψ1α i α tions are equivalent to simultaneous rotation of both vec- ψ = 2α , i =1, 2, α =1, 2 , (14)  ψ  tors, while U(1) changes phases of all µa simultaneously, which is equivalent to SO(2) rotation inside each couple together with their complex conjugates, Re µa, Im µa . Only in case when these vectors are par- allel{ we have} an invariance of the mass matrix which is α˙ ψ (ψ1˙α)∗, (ψ2˙α)∗ , i =1, 2, α˙ =1, 2 , (15) just a rotation around this common direction. This sym- i ≡   metry is the one identified with the baryonic U(1)B in representing the right-handed spinors. One can raise and Eq. (2). When it happens all Im µa can be absorbed in Re µa by U(1) transformation. lower spinor α, α˙ and flavor i indices using ǫαβ , ǫα˙ β˙ and 12 Let us show now that in the absence of the common ǫik with ǫ = 1. In terms of Dirac spinor n two left-handed Weyl spinors direction we get two spin 1/2 Majorana fermions with c 2 ∗ different masses. From equations of motion (14) are associated with nL and nL = iγ (nR) . In par- ticular, in the chiral (Weyl) basis of gamma-matrices we iα ik i ∂αα˙ ψ m ψkα˙ =0 , − (19) i ∂αα˙ ψ m ψkα =0 , iα˙ − ik 2 µ we come to the eigenvalue problem for M = pµp , 1 See, e.g., the book [14] where the Weyl spinor formalism is grace- fully applied to description of massive neutrinos. M 2ψkα mkim ψlα =0 . (20) − il 4

Using definition (18) of µa the squared mass matrix can discrete symmetries in this basis. be presented as a combination of isoscalar and isovector pieces: 5. In the Weyl description with the mass matrix m0 kl a a k abc a b c k given by (25) the charge conjugation C, m mln = µ µ δn + iǫ µ µ (τ )n . (21) b 1 α 2 α 2 C : ψ ψ , ψ ψ , (27) Correspondingly, there are two invariants defining M . ←→ 1˙α ←→ 2˙α The isoscalar part gives the sum of eigenvalues, is just interchanging fields of the same chirality but with M 2 + M 2 the opposite baryon charges. In terms of U(2) transfor- 1 2 = µa µa = (Re µa)2 +(Im µa)2 (22) 2 mations it is the SU(2) rotation by angle π around the first axis up to the factor ( i) which is the U(1) rotation: while the length of the isovector part defines the splitting − 1 of the eigenvalues, C : ψ U ψ, U =e−iπ/2eiπτ /2 = τ 1 . (28) → C C 2 2 2 M1 M2 abc a b This is in the basis where the mass matrix has the quasi- − =2 ǫ Re µ Im µ . (23) 3 2 rh i Dirac form (25). For generic form of the mass matrix we can use Eq.(26) to get This shows how the splitting is associated with the break- ing of the baryon charge. 1 † UC = V τ V . (29) To follow the discrete symmetries we can orient the a mass matrix mik in a convenient way. In terms of µ the Moreover, we can write the matrix UC in an arbitrary mass matrix m has the form basis,

1 2 3 a a a a µ iµ µ UC = exp( iπ/2) exp(iπτ n /2) = n τ , m =bmik = − −3 1 2 . (24)  µ µ iµ  − − ǫabc Im µb Re µc (30) b na = , Without lost of generality one can render the vectors ǫabc Im µb Re µc Re µa and Im µa orthogonal using the overall U(1) phase transformation. Then by remaining SU(2) rotations we with a straightforward geometrical interpretation. In- can put both of them onto the 23 plane, i.e., put µ1 = 0. deed, it is just a combination of the SU(2) rotation and choose the direction of Im µa as the 2-nd axis. So, around the normal na to the plane of Re µa and Im µa by only two non-vanishing parameters, Re µ3 and Im µ2, re- angle π with the chiral U(1) rotation by angle ( π/2). main and the mass matrix takes the form Evidently, this is a discrete symmetry of the mass− ma- T Im µ2 Re µ3 ǫ m trix, UC mUC = m . The SU(2) rotation by π changes m0 = 3 2 = , (25) the sign of m and the U(1) rotation, exp( iπ/2) = i ,  Re µ Im µ   m ǫ  compensatesb thisb sign. − − b 2 where correspondence, m=Re µ3, ǫ=Im µ2. with param- The transformationb C together with C = I composes eters introduced earlier in four-component spinor nota- the discrete Z2 subgroup that survives from U(2) for a 2 2 generic mass term. The only other discrete symmetry is tions is also shown. Then M1,2 = (m ǫ) as in the previous Section. ± Z2 associated with changing sign for all fermion fields. P In other words, an arbitrary mass matrix m, as in Let us turn now to the parity transformation z de- Eq.(24), can be brought to quasi-Dirac form m0, given fined by by Eq.(13) in terms of Dirac spinors. In terms by Eq.(25) with real parameters m and ǫ, by ab certain of Weyl spinors (14) and (15) the inversion of space co- U(2) transformation V , b ordinates then implies T P 1α 2α m0 = V mV . (26) z : ψ i ψ2˙α , ψ i ψ1˙α, → → (31) 2α 1α Indeed, 6 real parameters in the matrix m are diminished ψ1˙α iψ , ψ2˙α iψ . b b → → to 2 in m0 by 4 parameters of U(2) rotations. In the limit ǫ = 0 the neutron becomesb a Dirac This is in the basis where the mass matrix has the form (25). Again, similar to C, it can be written in the form: particle,b and the baryon symmetry U(1)B associated with SU(2) rotations around 3-rd axis with a diagonal † P : ψ i ψU , ψ iU ψ , (32) generator τ 3/2 arises. Non-zero Majorana mass ǫ breaks z → P → P this symmetry but in real situation ǫ m, the neutron ≪ behaves practically as Dirac particle, and U(1)B remains 2 an approximate symmetry. It is convenient to discuss 3 In the limit ǫ = 0, when only 3-rd axis is fixed, m = Re µ3, one can consider any combination of rotations around 1-st and 1 2 2-nd axes, UC = cos ω τ + sin ω τ . This is the origin of the well-known phase freedom in the definition of C transformation 2 Present experimental limits on n−n¯ oscillation [3] yield the upper for Dirac fermion, n → e−iωnc, nc → eiωn. Non-zero ǫ removes bound ǫ< 2.5 × 10−33 GeV. the phase freedom and leaves only the possibility sin ω = 0. 5

1 1 T where UP = τ in the basis (25) and UP = V τ V in an ratios of “wrong” decays between the neutron and arbitrary basis. The transformation (32) clearly demon- antineutron, Br(n pe¯ +ν) = Br(¯n pe+ν¯), even if strates P2 = 1.4 observation of these→ decays is only6 a gedanken→ possibility. z − The operation CPz which changes both, charge and However, some CPz violating processes related to new chirality, has the form, -violating physics that induces the n n¯ oscillation can Bbe at the origin of the baryon asymmetry− of the Universe. CP : ψ i ψU †U =i ψV V T , ψ U †U ψ =iV ∗V †ψ . z → C P → P C (33) 6. In the Standard Model (SM) conservations of baryon kα kα It is just ψ i ψkα˙ and ψkα˙ iψ in the basis (25). and lepton numbers are related to accidental global C → P → CP 2 Note, that and z commute and ( z) = 1. symmetriesB ofL the SM Lagrangian.5 The violation of by T − Finally, one can define transformation which besides two units can originate only from new physics beyondB SM the time inversion and reordering operators in the La- which would induce the effective six-quark interaction grangian (17) implies 1 i T iα iα (∆ = 2) = 5 ci , : ψ ψiα˙ , ψiα˙ ψ , (34) L B − M O (37) → →− X i = T i qA1 qA2 qA3 qA4 qA5 qA6 , in the basis (25) and O A1A2A3A4A5A6 where coefficients T i account for different flavor, color T: ψ ψV V T , ψ V ∗V †ψiα , (35) → →− and spinor structures and the large mass scale M com- T ing from new physics leads to the smallness of baryon in the arbitrary basis. Clearly, anti-commutes with violation. CP and T2 = 1. z − In particular, the nn¯ mixing term (5) emerges as a Combining, we get CPzT transformation, which acts 5 matrix element between n andn ¯ states of the operator as n iγ n on the Dirac spinor together with inversion (37), see diagram in Fig. 1, of all→ space-time coordinates and reordering of operators in the Lagrangian, 1 n¯ (∆ = 2) n = ǫ vT Cu , (38) h | L B − | i −2 n¯ n CP T 1α iα z : ψ iψ , ψiα i ψiα . (36) → →− where un, vn¯ are Dirac spinors for n,n ¯. Generically, it 2 gives a complex value for ǫ but by a phase redefinition It satisfies (CPzT) = 1 and presents an invariance of any local and Lorentz-invariant− Lagrangian. of n, n¯ states we always can make it real and positive. Concluding this section, let us emphasize that we have Thus, an estimate of the parameter ǫ, which is inverse of shown that for any pattern of the neutron mass terms, the oscillation time τnn¯ , is including the Dirac mass respecting the baryon number 6 1 ΛQCD conservation, as well as the Majorana ones violating it ǫ = 5 . (39) by two units, one can always consistently define the op- τnn¯ ∼ M erations of parity transformation Pz and charge conju- gation C as preserved symmetries in spite of breaking of For u and d quarks of the first generation the full list of ∆ = 2 six-quark operators was determined in the baryon charge conservation. In fact, a generic mass B − matrix m in (24) can be always rotated by flavor transfor- Refs. [15, 16], T mation V mV to a pseudo-Dirac form (25) where these 1 = uiT Cuj d kT Cd l d mT Cd n ǫ ǫ + symmetriesb are defined in an unique way. Oχ1χ2χ3 χ1 χ1 χ2 χ2 χ3 χ3 ikm jln  Thus, theb neutron-antineutron oscillation in itself ǫiknǫjlm + ǫjkmǫnil + ǫjknǫilm , C does not violate discrete symmetries. However, , 2 iT j kT l mT n  P CP T χ1χ2χ3 = uχ1 Cdχ1 uχ2 Cdχ2 dχ3 Cdχ3 ǫikmǫjln+ z and also z (which is an equivalent of ), O  (40) generically will not be respected by the interaction ǫiknǫjlm + ǫjkmǫnil + ǫjknǫilm , terms. Consider, e.g., the neutron β-decay n peν¯,  → 3 = uiT Cdj u kT Cd l d mT Cd n ǫ ǫ + implying that interaction has the standard, baryon Oχ1χ2χ3 χ1 χ1 χ2 χ2 χ3 χ3 ijm kln  charge preserving, form. Then the presence of 6B ǫijnǫklm . L+ terms would induce also the “wrong” decays n pe¯ ν  → Here χ stands for L or R quark chirality. Accounting (though extremely suppressed). Furthermore, CPz i violation could be manifested in difference of branching for relations 1 1 2,3 2,3 χLR = χRL , LRχ = RLχ , O O O O (41) 2 1 3 χχχ′ χχχ′ =3 χχχ′ , 4 In the limit ǫ = 0, Pz transformation can be combined with O − O O U(1)B rotations. In particular, P transformation (11) is a com- 3 3 bination of Pz and a discrete baryon rotation iτ = exp(iπτ /2), so that P2 = 1 and CP = −PC. Once again, such a definition 5 Nonperturbative breaking of B and L, preserving B−L, is ex- of parity makes sense only for a Dirac fermion. tremely small. 6

Again, our phase definitions for quarks are consistent with those for neutron. So, combinations Oi H.c. (43) χ1χ2χ3 ± represent C even and C odd operators. In total, we break all 28 operators into four groups with different Pz, C and CPz features, each group contains seven operators, d d Oi +L R +H.c., P =+ , C =+ , CP = + ; u u χ1χ2χ3 ↔ z z { {  i  P C CP d d O +L R H.c., =+ , = , = ; ݊ ෤݊ χ1χ2χ3 ↔ − z − z −   Oi L R +H.c., P = , C =+ , CP = ; χ1χ2χ3− ↔ z − z −   Oi L R H.c., P = , C = , CP =+ . χ1χ2χ3− ↔ − z − − z   (44)

Only the first seven operators, which are both Pz and C even, contribute to nn¯ oscillations. It is, of course, up to small corrections due to electroweak interactions where the discrete symmetries are broken. What about the remaining 21 combinations which are FIG. 1. Diagram for generating n − n¯ mixing terms odd either under Pz or C transformations? Although they do not contribute to the n n¯ transition, their effect show up in instability of nuclei.− This source of instability in this case is not due to neutron-antineutron oscillations but due to processes of annihilation of two nucleons inside nucleus like N + N π + π, and, in particular, two d d proton annihilation, pp→ π+π+, shown on Fig. 2. This ା ା could be particularly interesting→ in case of suppressed nn¯ ߨ ߨ .uݑ ݑu oscillations The operators of the type of (37) involving strange u u quark, like udsuds, could induce Λ Λ¯ mixing. However, { − {d d ݌ such operators would also lead to nuclear instability via ݌ nucleon annihilation into kaons N + N K + K, see the diagram in Fig. 2 where in upper lines→ d quark is substituted by s quark (and π+ by K+). In fact, nuclear instability bounds on Λ Λ¯ mixing are only mildly, within an order of magnitude,− weaker than with respect to n n¯ mixing which makes hopeless the possibility to detect− Λ Λ¯ oscillation in the hyperon beam. (Instead, it can be− of interest to search for the nuclear decays into kaons in the large volume detectors.) The nuclear + + instability limits on Λ Λ¯ mixing are about 15 orders of FIG. 2. Inducing pp → π π annihilation via operators (37) − −6 magnitude stronger than the sensitivity δΛΛ¯ 10 eV which can be achieved in the laboratory∼ conditions [17]. The nuclear stability limits make hopeless also the we deal with 14 operators for ∆ = 2 transitions and laboratory search of bus-like baryon oscillation due to 14 Hermitian conjugated ones forB ∆ −= +2. operator usbusb suggested in Ref. [18]. B The Pz reflection interchanges L and R chirality χi i P in the operators Oχ1χ2χ3 . Note, that the z reflection 7. Our above consideration refers to the neutron-anti- for u and d quarks is defined similar to the neutron by neutron oscillation in vacuum. Now we show that even in Eq. (13). This is consistent with the udd wave function the presence of magnetic field no new ∆ = 2 operator P of neutron. Thus, we can divide operators into z even appears. A similar consideration was done| B| in Ref. [19] in P and z odd ones, application to a possible magnetic moment of neutrino. i In the Weyl formalism the field strengths tensor Fµν Oχ1χ2χ3 L R. (42) ± ↔ is substituted by the symmetric tensor Fαβ and its com- C ~ ~ The charge conjugation transforms operators plex conjugate Fα˙ β˙ . They correspond to E iB com- i i † ± Oχ1χ2χ3 into the Hermitian conjugated [Oχ1 χ2χ3 ] . binations of electric and magnetic fields. Then Lorentz 7

µν 5 ′ invariance allows only two structures involving electro- nσ γ n Fµν are allowed which describe respectively magnetic fields, the transitional magnetic and electric dipole moments ′ ˙ between n and n states (and the similar operators iα kβ ¯α˙ ¯β ik ′ Fαβ ψ ψ ǫik , Fα˙ β˙ ψi ψk ǫ (45) with mirror electromagnetic field, Fµν Fµν ). Since underlying new physics generating n n′→mixings gener- Antisymmetry in flavor indices implies that spinors with ically should violate CP-invariance,− both transitional the opposite baryon charges enter. So both operators magnetic and electric dipole moments can be of the preserve the baryon charge, and in fact they describe in- same order. In large enough magnetic (or electric) teractions with the magnetic and electric dipole moments field n n′ transition probabilities should not depend of the neutron. In terms of Majorana mass eigenstates on the→ field value, with possible implications for the (9), these are transitional moments between n1 and n2. search of neutron mirror neutron transitions, and in However, no transitional moment can exist between n − c particular for testing experimentally solution of the and n . neutron lifetime puzzle via n n′ transitions [22]. The authors of Ref.[20] realize that the operator − nT σµν CnF with ∆ = 2 is vanishing due to Fermi µν 8. statistics. They believe,B however,− that a composite na- Our use of the effective Lagrangian for the proof CPT ture of neutron changes the situation and a new type of means that the Lorentz invariance and are cru- magnetic moment in ∆ = 2 transitions may present. cial inputs. Once constraints of Lorentz invariance are B ± lifted new ∆ = 2 operators could show up. In other words, they think that the effective Lagrangian | B| description is broken for composite particles. Such operators were analyzed in Ref. [23] for putting To show that is not the case let us consider the process limits on the Lorentz invariance breaking. In particular, T 5 2 of annihilation of two neutrons into virtual photon, the authors suggested the operator n Cγ γ n as an ex- ample which involves spin flip and, correspondingly, less ∗ n(p1)+ n(p2) γ (k) , (46) dependent on magnetic field surrounding. → Note, however, that besides breaking of Lorentz which is the crossing channel to n nγ¯ ∗ transition. The − invariance this operator breaks also 3d rotational invari- number of invariant amplitudes for the process (46) which ance, i.e., isotropy of space. Such anisotropy could be is 1/2+ +1/2+ 1− transition is equal to one. Only → studied by measuring spin effects in neutron-antineutron orbital momentum L = 1 and total spin S = 1 in the transitions. two neutron system are allowed by angular momentum conservation and Fermi statistics. The gauge-invariant 9. form of the amplitude is The construction we used for neutron-antineutron transition could be applied to mixing of massive neutri- T µ ν u (p1)Cγ γ5u(p2) k k ǫ k ǫ , (47) nos. As an example, let us take the system of left-handed µ ν − ν µ
νe and νµ and their conjugated partners, right-handed where u1 2 are Dirac spinors describing neutrons and ǫ , µ ν¯e andν ¯µ. One can ascribe them [24] a flavor charge refers to the gauge potential. In space representation we = e µ (analog of ), to be (+1) for νe and (-1) deal with ∂ν F the quantity which vanishes outside of F L − L B µν for νµ. Then, C conjugation is interchange of νe and νµ. the source of the electromagnetic field, and, in particular, Again, breaking mass term would be C and Pz even for the distributed magnetic field. It proves that there is but oddF for P. no place for magnetic moment of n n¯ transition, and − A similar scenario can be staged in case of Dirac effective Lagrangian description does work. Let us also massive neutrino. remark that n nγ¯ ∗ transition with a virtual photon connected to the→ proton, as well as nn γ∗ annihilation, 10. CPT would destabilise the nuclei even in the→ absence of n n¯ In summary, we show that the Lorentz and − invariance lead to the unique ∆ = 2 operator in the mass mixing. | B| Even in the absence of new n n¯ magnetic moment the effective Lagrangian for the neutron-antineutron mixing. C authors of [20] claim that suppression− of n n¯ oscillations This mixing is even under the charge conjugation as P by external magnetic field can be overcome− by applying well as under the modified parity z which takes the same the magnetic field transversal to quantization axis. Fol- value i for both, neutron and antineutron in contrast with standard (+1) and ( 1) values. It means that observa- lowing our criticism [4] Gardner and Yan recognized in − [7] that it would break the rotational invariance. As a tion of the neutron-antineutron mixing per se does not give a signal of CP violation. It could be compared with consequence the magnetic field suppression does present 0 indeed. the K0 K transition amplitude with ∆S = 2 where − | | The situation is different if one considers oscillation to separate CP conserving and CP breaking parts one n n′ where n′ is a mirror neutron, twin of the neutron needs to relate it to ∆S = 1 decay amplitudes. | | from− hidden mirror sector [21]. In this case one deals We applied the discrete symmetries to classification with the mass mixing between two Dirac fermions, of possible ∆ = 2 six-quark operators, separating εnn′ + h.c., conserving a combination of baryon num- those which| contributesB| to the neutron-antineutron mix- bers + ′. Hence, also operators nσµν n′F and ing. Other ∆ = 2 operators contribute to instablity of B B µν | B| 8 nuclei. We also showed that switching on external magnetic We thank Martin Einhorn, Susan Gardner, Yuri field influences the level splitting, what suppresses n n¯ Kamyshkov, Kirill Melnikov, Rabi Mohapatra, Adam − oscillations, but does not add any new ∆ = 2 operator Ritz and Misha Voloshin for helpful discussions. A.V. | B| in contradistinction with recent claims in literature. appreciates hospitality of the Kavli Institute for Theoret- Our classification of ∆ = 2 operators coming from ical Physics where his research was supported in part by | B| new physics, could be useful in association with Sakharov the National Science Foundation under Grant No. NSF conditions for baryogenesis which involves both, non- PHY11-25915. conservation of baryon charge and CP-violation.

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